Theorem ssintab

Description: Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006) (Proof shortened by Andrew Salmon, 9-Jul-2011)

	Ref	Expression
Assertion	ssintab	⊢ ( 𝐴 ⊆ ∩ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝜑 → 𝐴 ⊆ 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		ssint	⊢ ( 𝐴 ⊆ ∩ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑦 ∈ { 𝑥 ∣ 𝜑 } 𝐴 ⊆ 𝑦 )
2		sseq2	⊢ ( 𝑦 = 𝑥 → ( 𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑥 ) )
3	2	ralab2	⊢ ( ∀ 𝑦 ∈ { 𝑥 ∣ 𝜑 } 𝐴 ⊆ 𝑦 ↔ ∀ 𝑥 ( 𝜑 → 𝐴 ⊆ 𝑥 ) )
4	1 3	bitri	⊢ ( 𝐴 ⊆ ∩ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝜑 → 𝐴 ⊆ 𝑥 ) )